Dirichlet series associated to quartic fields with given cubic resolvent

Henri Cohen,Frank Thorne

doi:10.1007/s40993-015-0001-y

Abstract

Let $k$ be a cubic field. We give an explicit formula for the Dirichlet series $\sum_K|\Disc(K)|^{-s}$, where the sum is over isomorphism classes of all quartic fields whose cubic resolvent field is isomorphic to $k$. Our work is a sequel to an unpublished preprint of Cohen, Diaz y Diaz, and Olivier, and we include complete proofs of their results so as not to rely on unpublished work. This is a companion to a previous paper where we compute the Dirichlet series associated to cubic fields having a given quadratic resolvent.

Highlights

1 Background In a previous paper [14], we studied the problem of enumerating cubic fields1 with fixed quadratic resolvent
We have Disc(K) = Disc(k)f (K)2 for a positive integer f (K), and for each fixed k we proved explicit formulas for the Dirichlet series K f (K)−s,√where the sum is over all cubic fields K with quadratic resolvent k
9 Results with signatures In the case of cubic fields with given quadratic resolvent, the quadratic resolvent determines the signature of the cubic field

Summary

Introduction

In a previous paper [14], we studied the problem of enumerating cubic fields1 with fixed quadratic resolvent. It is fundamental to our efforts that quartic fields K with cubic resolvent k correspond to quadratic extensions K6/k of trivial norm.

Results

Conclusion